Key Concepts of Parametric Surfaces

Why This Matters

Parametric surfaces are your gateway to describing and analyzing complex three-dimensional shapes that would be impossible to capture with a single equation like $z = f(x, y)$. In Calc III, you're being tested on your ability to set up parametrizations, compute tangent planes and normal vectors, and evaluate surface integrals—all of which depend on understanding how two parameters map to 3D space. These concepts connect directly to vector calculus applications like flux, surface area, and orientation.

Don't just memorize the formulas for spheres and cylinders—know why each parametrization works and how the partial derivatives generate the geometric information you need. Exam questions will ask you to parametrize unfamiliar surfaces, find normal vectors, or set up surface integrals, so understanding the underlying mechanics is what separates a 5 from a 3.

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Foundations: What Parametric Surfaces Are

Before diving into specific surfaces, you need to understand the core machinery. A parametric surface maps a 2D parameter domain to 3D space, letting you trace out every point on a surface systematically.

Definition of Parametric Surfaces

• Vector function $\mathbf{r}(u, v)$—maps two parameters to a position vector $(x(u,v), y(u,v), z(u,v))$ in three-dimensional space
• Parameter domain $D$ is typically a rectangle or region in the $uv$-plane that gets "stretched" onto the surface
• Key advantage: allows representation of surfaces that can't be written as $z = f(x,y)$, like spheres or Möbius strips

Partial Derivatives of Parametric Surfaces

• $\mathbf{r}_u$ and $\mathbf{r}_v$ are tangent vectors pointing in the direction of increasing $u$ and $v$ respectively
• These derivatives are the building blocks for everything else: tangent planes, normal vectors, and surface area all derive from them
• Computed component-wise: if $\mathbf{r}(u,v) = (x, y, z)$, then $\mathbf{r}_u = \left(\frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u}\right)$

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Standard Surface Parametrizations

These are the surfaces you'll see repeatedly on exams. Each uses a different coordinate system or geometric insight to map two parameters onto the shape.

Parametrization of a Plane

• Formula: $\mathbf{r}(u, v) = \mathbf{p} + u\mathbf{a} + v\mathbf{b}$—where $\mathbf{p}$ is a point on the plane and $\mathbf{a}, \mathbf{b}$ are direction vectors
• Direction vectors $\mathbf{a}$ and $\mathbf{b}$ must be non-parallel; they determine the plane's orientation in space
• Simplest parametric surface: partial derivatives are constant ($\mathbf{r}_u = \mathbf{a}$, $\mathbf{r}_v = \mathbf{b}$), making calculations straightforward

Parametrization of a Sphere

• Spherical coordinates: $\mathbf{r}(\theta, \phi) = (R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi)$ with radius $R$
• Parameter bounds: $\theta \in [0, 2\pi)$ sweeps around the equator, $\phi \in [0, \pi]$ sweeps from north pole to south pole
• Watch for singularities at the poles where $\phi = 0$ or $\phi = \pi$—the tangent vector $\mathbf{r}_\theta$ becomes zero there

Parametrization of a Cylinder

• Formula: $\mathbf{r}(u, v) = (R\cos u, R\sin u, v)$—$u$ wraps around the circle, $v$ moves up/down the axis
• Parameter bounds: $u \in [0, 2\pi]$ for a full revolution, $v$ ranges over the desired height
• Constant radius $R$ distinguishes this from a cone; the cross-section is the same circle at every height

Parametrization of a Cone

• Formula: $\mathbf{r}(u, v) = (v\cos u, v\sin u, h - kv)$—radius grows linearly with $v$
• Parameter $v$ controls both the radial distance from the axis and the height, creating the sloped surface
• Constant $k$ determines the cone's steepness; larger $k$ means a wider cone

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Geometric Analysis Tools

Once you have a parametrization, these tools let you extract geometric information. The cross product of partial derivatives is the central operation here.

Tangent Planes to Parametric Surfaces

• Formula: tangent plane at $(u_0, v_0)$ is $\mathbf{r}(u_0, v_0) + s\mathbf{r}_u(u_0, v_0) + t\mathbf{r}_v(u_0, v_0)$ for scalars $s, t$
• Requires both partial derivatives evaluated at the point of tangency—these span the tangent plane
• Local linear approximation: the tangent plane is the best flat approximation to the surface near that point

Normal Vectors to Parametric Surfaces

• Cross product formula: $\mathbf{N} = \mathbf{r}_u \times \mathbf{r}_v$—perpendicular to both tangent vectors, hence perpendicular to the surface
• Direction matters: the order of the cross product determines which way $\mathbf{N}$ points (outward vs. inward)
• Applications include surface integrals, flux calculations, and determining how light reflects off surfaces

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Surface Integrals and Orientation

These concepts connect parametric surfaces to integral calculus. Orientation determines the sign of your answer in flux integrals.

Surface Area of Parametric Surfaces

• Formula: $A = \iint_D \|\mathbf{r}_u \times \mathbf{r}_v\| \, du \, dv$—the magnitude of the cross product measures local stretching
• $\|\mathbf{r}_u \times \mathbf{r}_v\|$ is the area element that accounts for how the flat parameter domain gets distorted onto the curved surface
• Critical for applications: surface area appears in physics (heat transfer, material costs) and sets up surface integrals of scalar functions

Orientation of Parametric Surfaces

• Orientation = choice of normal direction—determined by the order of parameters in $\mathbf{r}_u \times \mathbf{r}_v$
• Swapping $u$ and $v$ reverses the normal vector direction (since $\mathbf{r}_v \times \mathbf{r}_u = -\mathbf{r}_u \times \mathbf{r}_v$)
• Essential for flux integrals: getting orientation wrong gives you the negative of the correct answer

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Quick Reference Table

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Self-Check Questions

1. What do the sphere and cylinder parametrizations have in common, and how do their parameter bounds differ?

2. Given a parametric surface $\mathbf{r}(u,v)$, what two quantities do you need to compute before finding the surface area, and how do you combine them?

3. If you switch the order of parameters from $\mathbf{r}_u \times \mathbf{r}_v$ to $\mathbf{r}_v \times \mathbf{r}_u$, what happens to your normal vector and why does this matter for flux integrals?

4. Compare and contrast the parametrization of a cone versus a cylinder—what single change transforms one into the other?

5. (FRQ-style) A surface is given by $\mathbf{r}(u,v) = (u\cos v, u\sin v, u^2)$. Describe the surface, find the normal vector at a general point, and set up (but don't evaluate) the integral for surface area over $0 \leq u \leq 1$, $0 \leq v \leq 2\pi$.